Israeli war cabinet member Benny Gantz has warned that unless Hamas frees all hostages held in Gaza by 10 March an offensive will be launched in Rafah, reports BBC.
It is the first time Israel has said when its troops might enter Gaza's overcrowded southern city Rafah.
Global opposition is growing to such an attack in Rafah, where some 1.5 million Palestinians are sheltering.
Earlier, the UN public health agency said a key Gaza hospital had ceased to function following an Israeli raid.
On the other hand, the World Health Organization (WHO) said it had not been allowed to enter Nasser hospital in Khan Yunis, north of Rafah, to assess the situation.
The Israel Defence Forces (IDF) entered the complex on Thursday, saying intelligence indicated hostages taken by Hamas were being held there.
The IDF has described its operation in Nasser as "precise and limited", accusing Hamas of "cynically using hospitals for terror".
Speaking on Sunday, Gantz, a former defence minister, said, "The world must know, and Hamas leaders must know - if by Ramadan our hostages are not home, the fighting will continue everywhere, to include the Rafah area".
Ramadan, the Islamic holy month of fasting - this year begins on 10 March.
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